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Mirrors > Home > ILE Home > Th. List > tgidm | Unicode version |
Description: The topology generator function is idempotent. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
tgidm |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgvalex 12779 |
. . . . 5
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2 | eltg3 14042 |
. . . . 5
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3 | 1, 2 | syl 14 |
. . . 4
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4 | uniiun 3958 |
. . . . . . . . . 10
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5 | simpr 110 |
. . . . . . . . . . . . 13
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6 | 5 | sselda 3170 |
. . . . . . . . . . . 12
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7 | eltg4i 14040 |
. . . . . . . . . . . 12
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8 | 6, 7 | syl 14 |
. . . . . . . . . . 11
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9 | 8 | iuneq2dv 3925 |
. . . . . . . . . 10
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10 | 4, 9 | eqtrid 2234 |
. . . . . . . . 9
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11 | iuncom4 3911 |
. . . . . . . . 9
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12 | 10, 11 | eqtrdi 2238 |
. . . . . . . 8
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13 | inss1 3370 |
. . . . . . . . . . . 12
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14 | 13 | rgenw 2545 |
. . . . . . . . . . 11
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15 | iunss 3945 |
. . . . . . . . . . 11
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16 | 14, 15 | mpbir 146 |
. . . . . . . . . 10
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17 | 16 | a1i 9 |
. . . . . . . . 9
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18 | eltg3i 14041 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 17, 18 | sylan2 286 |
. . . . . . . 8
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20 | 12, 19 | eqeltrd 2266 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | eleq1 2252 |
. . . . . . 7
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22 | 20, 21 | syl5ibrcom 157 |
. . . . . 6
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23 | 22 | expimpd 363 |
. . . . 5
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24 | 23 | exlimdv 1830 |
. . . 4
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25 | 3, 24 | sylbid 150 |
. . 3
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26 | 25 | ssrdv 3176 |
. 2
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27 | bastg 14046 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | tgss 14048 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
29 | 1, 27, 28 | syl2anc 411 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 26, 29 | eqssd 3187 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 ax-un 4454 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-iun 3906 df-br 4022 df-opab 4083 df-mpt 4084 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-iota 5199 df-fun 5240 df-fv 5246 df-topgen 12776 |
This theorem is referenced by: tgss3 14063 txbasval 14252 |
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